Rational sequence; primes vs. odd numbers

[Andrew Plewe]

s1, the sum of the numerator and the denominator of the fractions are  
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Subject: Re: Rational sequence; primes vs. odd numbers
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Andrew Plewe wrote:
> All of the resulting values will be rational, but which values will have a
> finite decimal expansion? Doing some quick by-hand computations, I find:
> 
> (1/2 + ... + 14/15) - (1/2 + ... + 9/10) = 7/40 = 0.175, or the 9th term.
> 
> But that's the only one so far. Are there others? Are there infinitely many?
> 

In order for the decimal expansion to be finite, the two sums must have the
same set of primes (other than 2 and 5) in their common denominator.  (Note
that this condition is necessary but not sufficient.)  At the 9th term,
both sets of sums contain only the primes 2, 3, 5, and 7 in their
denominators, so a finite decimal expansion is possible, and in fact happens.

But after the 9th term, it looks like you never have the same set of primes
in the two denominators.  At the 10th term, you get an 11 in the denominator
of s1 which is not matched in s2.  At the 40th and 41st terms, it comes close,
as only the prime 79 causes a mismatch (79 is in the denominator of s2 but
not of s1).

Through the first 1000 terms, it never again comes that close to a matched
set, and at 1000 terms, the number of mismatched primes is around 100, with
no realistic hope of seeing zero again.